# American Institute of Mathematical Sciences

August  2019, 13(4): 755-786. doi: 10.3934/ipi.2019035

## Total generalized variation regularization in data assimilation for Burgers' equation

 1 Research Center of Mathematical Modelling (MODEMAT), Escuela Politécnica Nacional, Quito, Ecuador 2 Faculty of Mathematics, Professorship Numerical Mathematics (Partial Differential Equations), Chemnitz University of Technology, Chemnitz, Germany

This paper was developed within the Master Program in Mathematical Optimization at Escuela Politécnica Nacional de Ecuador

Received  May 2018 Revised  November 2018 Published  May 2019

We propose a second-order total generalized variation (TGV) regularization for the reconstruction of the initial condition in variational data assimilation problems. After showing the equivalence between TGV regularization and a Bayesian MAP estimator, we focus on the detailed study of the inviscid Burgers' data assimilation problem. Due to the difficult structure of the governing hyperbolic conservation law, we consider a discretize–then–optimize approach and rigorously derive a first-order optimality condition for the problem. For the numerical solution, we propose a globalized reduced Newton-type method together with a polynomial line-search strategy, and prove convergence of the algorithm to stationary points. The paper finishes with some numerical experiments where, among others, the performance of TGV–regularization compared to TV–regularization is tested.

Citation: Juan Carlos De los Reyes, Estefanía Loayza-Romero. Total generalized variation regularization in data assimilation for Burgers' equation. Inverse Problems & Imaging, 2019, 13 (4) : 755-786. doi: 10.3934/ipi.2019035
##### References:

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##### References:
Exact solution
SSIM index for the TV and TGV regularizations
Best solutions for TV (left) and TGV (right) regularizations
Observations vs Optimal state
Exact solutions and best solutions obtained for the experiment
Global convergence of the algorithm
Locally superlinear convergence of the algorithm
Exact solutions for the experiment
Results of the experiments with different values of $\mu$
 Experiment $\mu$ iter SSIM $J_\gamma$ time (s) 3 0 – NaN NaN – 1e-6 13 0.9594 35.8330 7.2 1e-8 12 0.9594 35.8330 6.9 1e-10 12 0.9594 35.8330 6.9 1e-12 13 0.9594 35.8333 7.1 1e-14 13 0.9594 35.8329 7.1 4 0 – NaN NaN – 1e-6 14 0.9592 39.2436 7.6 1e-8 13 0.9592 39.2434 7.0 1e-10 13 0.9592 39.2434 7.0 1e-12 14 0.9591 39.2437 7.7 1e-14 16 0.9591 39.2439 9.1
 Experiment $\mu$ iter SSIM $J_\gamma$ time (s) 3 0 – NaN NaN – 1e-6 13 0.9594 35.8330 7.2 1e-8 12 0.9594 35.8330 6.9 1e-10 12 0.9594 35.8330 6.9 1e-12 13 0.9594 35.8333 7.1 1e-14 13 0.9594 35.8329 7.1 4 0 – NaN NaN – 1e-6 14 0.9592 39.2436 7.6 1e-8 13 0.9592 39.2434 7.0 1e-10 13 0.9592 39.2434 7.0 1e-12 14 0.9591 39.2437 7.7 1e-14 16 0.9591 39.2439 9.1
Summary of the experiment
 Observations M N $\sigma$ 0.1 0.01 0.001 iter SSIM iter SSIM iter SSIM 4 75 40 20 0.9573 12 0.9895 12 0.9985 9 50 20 19 0.9576 12 0.9895 11 0.9985 30 25 10 20 0.9581 14 0.9896 11 0.9985 40 20 10 19 0.9584 15 0.9896 12 0.9985 100 15 5 15 0.9604 12 0.9896 11 0.9985 150 10 5 20 0.9617 13 0.9897 11 0.9985
 Observations M N $\sigma$ 0.1 0.01 0.001 iter SSIM iter SSIM iter SSIM 4 75 40 20 0.9573 12 0.9895 12 0.9985 9 50 20 19 0.9576 12 0.9895 11 0.9985 30 25 10 20 0.9581 14 0.9896 11 0.9985 40 20 10 19 0.9584 15 0.9896 12 0.9985 100 15 5 15 0.9604 12 0.9896 11 0.9985 150 10 5 20 0.9617 13 0.9897 11 0.9985
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